Method for computing factor of safety of a slope

ABSTRACT

The present application relates to method for computing the factor of safety of a slope including the following steps. Step 1: input parameters of a slope. Step 2: formulae a formula of stability number N into a target function and determine constrained conditions of geometric parameters α and λ. Step 3: obtain a minimum stability number N. Step 4: determine a position of a critical slip surface. Step 5: transform the critical slip surface defined by α, λ and n into a critical slip surface defined by R, x c  and y c , and compute overturning moments of slid masses above and below external water level of the critical slip surface. Step 6: resolve an equivalent unit weight. Step 7: resolve the factor of safety FS.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims a priority to Chinese Patent Application serialno. 202110462017.9, filed on Apr. 27, 2021. The entirety of theabove-mentioned patent application is hereby incorporated by referenceherein and made a part of this specification.

FIELD OF THE INVENTION

The present application relates to the field of slope engineering, andin particular, to a method for computing a factor of safety of a slope,especially a partially submerged undrained slope with strength linearlyincreasing and unlimited foundation depth.

DESCRIPTION OF RELATED ART

A slope chart is one of the methods for stability analysis, whichprovides an approach for rapidly analyzing the stability of a slope. Itcan be used for preliminary analysis and detailed analysis of checking,since it can provide an extremely fast result, especially being usefulfor selecting design schemes.

Taylor (1937) firstly developed a simple stability chart for evaluatingthe safety factor of homogeneous soil slopes based on a friction circlemethod. Such method attracted the attention of many scholars whocorrespondingly proposed a wide variety of stability charts foranalyzing 2-dimentional slope stability, so that they were widely usedin engineering practice.

Gibson & Morgenstern (1962) firstly proposed a stability chart on thestability of slope cut into normally consolidated and undrained claywhere the strength increases linearly with depth. In particular, theyestablished a relationship between factor of safety and slopeinclination for the case where the shear strength is zero at the groundsurface and increase linearly with depth, which means that the groundwater table is at or above the ground surface. Hunter & Schuster (1968)extended the analytical solution of Gibson & Morgenstern (1962) to thecase of ground water table below the ground surface. For this commoncase, according to the consolidation theory of clay, the shear strengths_(u0) at this time is greater than zero at the ground surface but stillincreases linearly with depth. By focusing on the theoretical researchon the stability of slope cut into normally consolidated and undrainedclay, Hunter & Schuster established an idealized strength-depthrelationship associated with a specific water lever h in the slope.

Neither of the above studies involves in the situation where water ispresent outside a slope. In actual practice, the stability chartprovided by Hunter and Schuster (1968) for a undrained slope where thestrength increases linearly with depth was further extended into thestability analysis of a slope with an external water level. However,such stability chart was established merely based on a slope withhomogeneous unit weight, so that they can be used in two situations,that is, a submerged slope and a slope without external water surface,but can not be directly applicable to a situation where a slope ispartially submerged.

Although stability analysis of an undrained slope, with or without anexternal water table, strictly involves total stresses, it has beennoted that the partially submerged undrained slope has the same factorof safety as a 2-layer soil slope, where the external water level hasbeen removed, and the soil in the slope above and beneath the originallocation of the outside water surface is set to its saturated andbuoyant unit weight, respectively.

Slope stability charts for layered soils typically require simplifiedproperties, so strategies have been proposed for determining averagestrength and unit weight that could be used as an equivalent“homogenized” property with charts. For example, for undrained clayslopes, it is usually considered sufficient to use a simple weightedaverage of the unit weight based on the thickness of each layer down tothe toe.

Using this approach for the 2-layer system, the average unit weight wasnot associated with a critical slip surface determined by a minimumstability number N while averaging the unit weight, a constantoverturning moment is not guaranteed, and the influence of a slope angleis not considered in these means. Instead, a constant average unitweight is obtained at a certain water level against different slopeangles, which, however, will results in a large error to the obtainedaverage unit weight, and even pass the error to the factor of safety.Further, the obtained factor of safety will be overestimated, andstability evaluation will probably be incorrect.

Therefore, there is still a need for a method capable of accuratelyevaluating the safety performance of a slope, especially a partiallysubmerged slope.

BRIEF SUMMARY OF THE INVENTION

In view of above problem, the present application provides a method forcomputing the factor of safety of a slope, especially a partiallysubmerged undrained slope, with a high accuracy.

In some embodiments, a method for computing the factor of safety of aslope includes the following steps.

Step 1: acquiring parameters of the slope including unit weights ofsoil, slope geometry, soil strength and external water level;

Step 2: formulating a formula of stability number N into a targetfunction and determining constrained conditions of variable geometricparameters in the target function;

Step 3: retrieving a combination of the variable geometric parameters αand λ by calling a genetic algorithm library through a Python program toobtain a minimum stability number N;

Step 4: determining a position of a critical slip surface according tothe retrieved combination of the variable geometric parameters and theslope geometry acquired in Step 1;

Step 5: transforming the determined critical slip surface into acritical slip surface defined by a circular arc radius R of the criticalslip surface and horizontal and vertical coordinate values x_(c) andy_(c) of a circle center of the critical slip surface, and computingoverturning moments of slid masses above and below external water levelof the critical slip surface;

Step 6: resolving an equivalent unit weight γ_(equiv) according tooverturning moment equilibrium; and

Step 7: resolving the factor of safety from the obtained minimumstability number N, a gradient ρ of soil strength and the equivalentunit weight obtained in Step 6 according to the following formula (1):

$\begin{matrix}{{{FS} = N^{\frac{\rho}{\gamma}}},} & (1)\end{matrix}$where γ is equal to the equivalent unit weight γ_(equiv).

In some other embodiments, a method for computing the factor of safetyof a slope includes the following steps.

Step 1: acquiring parameters of the slope including saturated unitweight of soil (γ_(sat)) and unit weight of water (γ_(w)), slopegeometry (β, H), soil strength (s_(u0), ρ, M) and external water level(d_(w)/H);

Step 2: formulating a formula of stability number N into a targetfunction and determining constrained conditions of geometric parametersα and λ;

Step 3: retrieving a combination of geometric parameters α and λ bycalling a genetic algorithm library through a Python program to obtain aminimum stability number N;

Step 4: determining a position of a critical slip surface according tothe obtained geometric parameters α, λ and dimensionless value n;

Step 5: transforming the critical slip surface defined by α, λ and ninto a critical slip surface defined by a circular arc radius R of thecritical slip surface and horizontal and vertical coordinate valuesx_(c) and y_(c) of a circle center of the critical slip surface, andcomputing overturning moments of slid masses above and below externalwater level of the critical slip surface;

Step 6: resolving an equivalent unit weight according to overturningmoment equilibrium; and

Step 7: resolving the factor of safety from the obtained minimumstability number N, a gradient ρ of soil strength and the equivalentunit weight obtained in Step 6 according to the following formula (1):

$\begin{matrix}{{{FS} = N^{\frac{\rho}{\gamma}}},} & (1)\end{matrix}$

In some embodiments, in Step 1, the unit weights include the saturatedunit weight (γ_(sat)) of the soil and the unit weight (γ_(w)) of waterwhile assigning the unit weight of soil above and below the externalwater level, saturated and buoyant unit weight, respectively. βrepresents a slope angle of the slope, and H represents a height of theslope. The undrained strength of soil is given by the equation:s _(u)(z)=s _(u0) +ρz  (2)where s_(u0) is the strength at crest level (z=0) and ρ is the gradientof strength linearly increasing with depth z. The external water levelparameter can be defined as d_(w)/H where d_(w) denotes a depth of waterlevel outside the slope from the top of the slope.

In some embodiments, in Step 2, the target function for a circular arcfailure mechanism can be expressed as:

$\begin{matrix}{N = {\frac{3}{{\sin^{2}\mspace{14mu}\alpha\mspace{11mu}\sin^{2}\mspace{14mu}\lambda}\;}\frac{\left\lbrack {{\cot\mspace{14mu}\lambda} + {\alpha\left( 1\rightarrow{{2M} - {\cot\mspace{14mu}\alpha\mspace{14mu}\cot\mspace{14mu}\lambda}} \right)}} \right\rbrack}{\begin{matrix}\left( {1 - {2\mspace{14mu}\cot^{2}\mspace{14mu}\beta} + {3\mspace{14mu}\cot\mspace{14mu}\lambda\mspace{14mu}\cot\mspace{14mu}\beta} +} \right. \\\left. {{3\mspace{14mu}\cot\mspace{14mu}\alpha\mspace{14mu}\cot\mspace{14mu}\lambda} - {3\mspace{14mu}\cot\mspace{14mu}\alpha\mspace{14mu}\cot\mspace{14mu}\beta}} \right)\end{matrix}}}} & (3)\end{matrix}$where α denotes half of a central angle of a circular arc slip surface,λ denotes an angle between a chord line of a circular arc slip surfaceand the horizontal plane, β denotes an angle of the slope, and M denotesa dimensionless strength gradient parameter. In some embodiments, M canbe defined as:

$\begin{matrix}{M = {\frac{h}{H}\frac{\gamma_{w}}{\gamma}}} & (4)\end{matrix}$where h denotes a specific water level in the slope, γ_(w) denotes aunit weight of water, and γ′ denotes a buoyant unit weight of soil. Inpractice, h can be neglected and thus M can be simplified as:

$\begin{matrix}{M = {\frac{H_{o}}{H} = \frac{s_{uo}}{\rho\; H}}} & (5)\end{matrix}$where H₀ denotes an intercept of a strength line.

In some other embodiments, in Step 2, the variable geometric parametersinclude α and λ, and the constrained conditions of variable geometricparameters α and λ can be, for example, α: [0, 90°] and λ: [0, β].

In some embodiments, in Step 4, the position of the critical slipsurface can be defined by

$\begin{matrix}{n = {\frac{1}{2}{\left( {{\cot\lambda} - {\cot\alpha} - {\cot\beta}} \right).}}} & (6)\end{matrix}$where the dimensionless value n denotes a measure of the distance thecritical slip surface outcrops beyond the toe of the slope; and, if n iszero or negative, the critical circles pass through the toe, as shown inFIG. 5B or FIG. 5C, and if n is positive, a critical circle exists belowthe toe, as shown in FIG. 5A.

In some embodiments, in Step 5, for a deep toe circle or a shallow toecircle, transforming the determined critical slip surface into acritical slip surface defined by R, x_(c) and y_(c) is performed by

$\begin{matrix}{R = \frac{H}{2\mspace{14mu}\sin\mspace{14mu}\alpha\mspace{14mu}\sin\mspace{14mu}\lambda}} & (7) \\{x_{c} = {\frac{H}{\tan\mspace{14mu}\beta} - {R\mspace{14mu}{\sin\left( {\alpha - \lambda} \right)}\mspace{14mu}{and}}}} & \left( {7\text{-}1} \right) \\{y_{c} = {{R\mspace{14mu}\cos*\left( {\alpha - \lambda} \right)} - {H.}}} & \left( {7\text{-}2} \right)\end{matrix}$

In some other embodiments, in Step 5, for a deep circle, transformingthe determined critical slip surface into a critical slip surfacedefined by R, x_(c) and y_(c) is performed by

$\begin{matrix}{R = \frac{H}{2\mspace{14mu}\sin\mspace{14mu}\alpha\mspace{14mu}\sin\mspace{14mu}\lambda}} & (7) \\{x_{c} = {\frac{H}{\tan\mspace{14mu}\beta} - {\left\lbrack {{R\mspace{14mu}{\sin\left( {\alpha - \lambda} \right)}} - {nH}} \right\rbrack\mspace{14mu}{and}}}} & \left( {7\text{-}3} \right) \\{y_{c} = {{R\mspace{14mu}\cos*\left( {\alpha - \lambda} \right)} - {H.}}} & \left( {7\text{-}4} \right)\end{matrix}$

In some embodiments, in Step 6, the overturning moment equilibrium isexpressed as:M ₁ +M ₂ =M ₀  (8)where M₀ denotes the total moment of a whole slip mass having anequivalent unit weight, M₁ denotes an overturning moment of saturatedunit weight of slip mass above the external water level, and M₂ denotesan overturning moment of buoyant unit weight of slip mass between theexternal water level and the toe of the slope.

In a further embodiment,M ₁=γ_(sat)∫_(−d) _(w) ⁰(x _(C) −x _(G))dA  (9)where γ_(sat) denotes saturated unit weight of soil, x_(G) denotes thehorizontal coordinate value of the centroid of a thin horizontalintegral element of soil at a general depth y, d_(w) denotes the depthof water outside slope measured below the crest, and dA denotes theintegral element area.

In a further embodiment,M ₂=γ′∫_(y) _(T) ^(−d) ^(w) (x _(C) −x _(G))dA  (10)where γ′ denotes buoyant unit weight of soil, and y_(T) denotes avertical coordinate of the toe of the slope.

In a preferred embodiment,M ₀=γ_(equiv)∫_(y) _(T) ⁰(x _(C) −x _(G))dA  (11).

In some embodiments, M₀ can be further expressed as:

$\begin{matrix}{M_{0} = {{\frac{\gamma\; H^{3}}{12}\left\lbrack {1 - {2\mspace{14mu}\cot^{2}\mspace{14mu}\beta} + {3\mspace{14mu}\cot\mspace{14mu}\lambda\mspace{14mu}\cot\mspace{14mu}\beta} + {3\mspace{14mu}\cot\mspace{14mu} a\mspace{14mu}\cot\mspace{14mu}\lambda} - {3\mspace{14mu}\cot\mspace{14mu}\alpha\mspace{14mu}\cot\mspace{14mu}\beta}} \right\rbrack}.}} & (12)\end{matrix}$

Then, the moment equilibrium can be further expressed as:

$\begin{matrix}\begin{matrix}{M_{0} = {{M_{1} + M_{2}} = {\gamma_{equiv}{\int_{\gamma_{T}}^{0}{\left( {x_{C} - x_{G}} \right){dA}}}}}} \\{= {\frac{\gamma\; H^{3}}{12}\left\lbrack {1 - {2\mspace{14mu}\cot^{2}\mspace{14mu}\beta} + {3\mspace{14mu}\cot\mspace{14mu}\lambda\mspace{14mu}\cot\mspace{14mu}\beta} +} \right.}} \\{\left. {{3\mspace{14mu}\cot\mspace{14mu} a\mspace{14mu}\cot\mspace{14mu}\lambda} - {3\mspace{14mu}\cot\mspace{14mu}\alpha\mspace{14mu}\cot\mspace{14mu}\beta}} \right\rbrack.}\end{matrix} & (13)\end{matrix}$

which reflects the equivalent relationship between the overturningmoment of partially submerged undrained slope and that of a uniformslope without external water surface and having equivalent unit weight.

In some embodiments, in order to resolve the equations (9), (10) and(11), from a circular arc equation of a slip surface:

$\begin{matrix}{{x = {x_{C} - \sqrt{R^{2} - \left( {y - y_{C}} \right)^{2}}}},{and}} & (14)\end{matrix}$a slope equation:

$\begin{matrix}{x = {- \frac{y}{\tan\mspace{20mu}\beta}}} & (15)\end{matrix}$it gives

$\begin{matrix}{x_{G} = {\frac{1}{2}\left( {x_{C} - \sqrt{R^{2} - \left( {y - x_{C}} \right)^{2}} - \frac{y}{\tan\mspace{20mu}\beta}} \right)}} & (16)\end{matrix}$which further gives

$\begin{matrix}{{x_{C} - x_{G}} = {\frac{1}{2}\left\lbrack {\sqrt{R^{2} - \left( {y - y_{C}} \right)^{2}} + \left( {\frac{y}{\tan\mspace{20mu}\beta} + x_{C}} \right)} \right\rbrack}} & (17) \\{{dA} = {\left\lbrack {\sqrt{R^{2} - \left( {y - y_{C}} \right)^{2}} - \left( {\frac{y}{\tan\mspace{20mu}\beta} + x_{C}} \right)} \right\rbrack{{dy}.}}} & (18)\end{matrix}$

Substituting equations (17) and (18) into equations (9)-(11) and theninto equation (13) provides an equivalent unit weight:

$\begin{matrix}{\gamma_{equiv} = {\frac{{\gamma_{sat}{\int_{- d_{W}}^{0}{\left( {x_{C} - x_{G}} \right){dA}}}} + {\gamma^{\prime}{\int_{\gamma_{T}}^{- d_{W}}{\left( {x_{C} - x_{G}} \right){dA}}}}}{\int_{\gamma_{T}}^{0}{\left( {x_{C} - x_{G}} \right){dA}}}.}} & (19)\end{matrix}$

In some embodiments, the equivalent unit weight γ_(equiv) can beobtained from an equivalent unit weight diagram represented by FIG. 14and equation (22).

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic view of an undrained slope with linearlyincreasing strength and an external water surface.

FIG. 2 is a schematic view of a slope equivalent to that in FIG. 1, withthe water removed and the buoyant unit weight substituted below wherethe water surface was previously.

FIG. 3 is a schematic view of simple averaging of unit weights in alayered undrained soil.

FIG. 4 is a schematic view of a slope equivalent to that in FIG. 1, withthe water removed and an equivalent unit weight used throughout.

FIGS. 5A-5C are three types of circular arc failure mechanisms of apartially submerged undrained slope: deep circle in FIG. 5A, deep toecircle in FIG. 5B, and shallow toe circle in FIG. 5C.

FIGS. 6A-6B are schematic views for computing an equivalent unit weightof a partially submerged undrained slope for a deep circle failuremechanism: a partially submerged undrained slope before homogenizing theunit weight in FIG. 6A and an equivalent slope without external watersurface after homogenizing the unit weight in FIG. 6B.

FIGS. 7A-7B are schematic views for computing an equivalent unit weightof a partially submerged undrained slope for a deep toe circle failuremechanism: a partially submerged undrained slope before homogenizing theunit weight in FIG. 7A and an equivalent slope without external watersurface after homogenizing the unit weight in FIG. 7B.

FIGS. 8A-8B are schematic views for computing an equivalent unit weightof a partially submerged undrained slope for a shallow toe circlefailure mechanism: a partially submerged undrained slope beforehomogenizing the unit weight in FIG. 8A and an equivalent slope withoutexternal water surface after homogenizing the unit weight in FIG. 8B.

FIG. 9 is a stability chart of an undrained slope (ϕ=0) where strengthincreases linearly with depth according to Hunter & Schuster (1968)(solid line) and after optimization in this application (dotted line).

FIG. 10 is a diagram of a factor of safety FS vs external water leveld_(w)/H at γ_(sat)=20 kN/m³, M=2, and β=30°.

FIG. 11 is a diagram of an equivalent unit weight γ_(equiv) fordifferent β and M at γ_(sat)=16 kN/m³ and d_(w)/H=0.5.

FIG. 12 is a diagram of γ_(equiv) vs β and d_(w)/H at γ_(sat)=16 kN/m³.

FIG. 13 is a diagram of FS vs β and d_(w)/H at γ_(sat)=16 kN/m³ and M=2.

FIG. 14 is a chart for η as a function of β and d_(w)/H whereγ_(equiv)=γ_(sat)−ηγ_(w) (solid curved lines), horizontal dotted linescorrespond to γ_(av) where η=1−d_(w)/H.

FIG. 15 is a schematic view of an example for computing the factor ofsafety of a partially submerged undrained slope (ϕ=0) where strengthincreases linearly with depth.

DETAILED DESCRIPTION

The analysis of equivalent unit weight of partially submerged undrainedslope with unlimited foundation depth where the strength increaseslinearly with depth is essentially an analysis of slope stability. Theaccuracy of solving the factor of safety by using a slope stabilitychart is usually determined by the accuracy of the shear strengthparameters as obtained. In addition, a slope failure or a land slidingis essentially a movement of rock-soil body down the slope under theaction of gravity. Therefore, the unit weight of soil has a significantimpact on the accuracy of the factor of safety FS.

The stability chart of an undrained slope where the strength increaseslinearly with depth was established regarding a simple slope havinghomogeneous unit weight, and thus can be directly applied to two cases,that is, a submerged case and a case without water outside the slope,but cannot be directly applied to a partially submerged undrained slope,as shown in FIG. 1.

Although stability analysis of an undrained slope, with or without anexternal water table, strictly involves total stresses, it has beennoted that the partially submerged undrained slope shown in FIG. 1 hasthe same factor of safety as a 2-layer soil slope shown in FIG. 2, wherethe water has been removed, and the soil in the slope above and beneaththe original location of the outside water surface is set to itssaturated and buoyant unit weight, respectively.

Slope stability charts for layered soils typically require simplifiedproperties, so strategies have been proposed for determining averagestrength and unit weight that could be used as an equivalent“homogenized” property with charts. For example, for undrained clayslopes, it is usually considered sufficient to use a simple weightedaverage of the unit weight based on the thickness of each layer down tothe toe. Using this approach for the 2-layer system shown in FIG. 3, theaverage unit weight would be given by equation (20)

$\begin{matrix}{\gamma_{av} = {\frac{{d_{w}\gamma_{sat}} + {\left( {H - d_{w}} \right)\gamma^{\prime}}}{H} = {\gamma_{sat} - {\left( {1 - \frac{d_{w}}{H}} \right)\gamma_{w}}}}} & (20)\end{matrix}$where γ_(av) is the average unit weight, γ_(sat) is the saturated unitweight of soil, γ_(w) is the unit weight of water, γ′ is the buoyantunit weight of soil below the external water surface, H is the height ofslope and d_(w) is the depth of water outside slope measured below thecrest.

Using this approach for the 2-layer system, the average unit weight wasnot associated with a critical slip surface determined by a minimumstability number N while averaging the unit weight, a constantoverturning moment is not guaranteed, and the influence of a slope angleis not considered in these means. Instead, a constant average unitweight is obtained at a certain water level against different slopeangles, which, however, will results in a large error to the obtainedaverage unit weight, and even pass the error to the factor of safety.Further, the obtained factor of safety will be overestimated, and theresult for stability evaluation will probably be incorrect.

The goal of this present application, therefore, is to find theequivalent unit weight as shown in FIG. 4 that gives the same factor ofsafety as the slopes in FIGS. 1 and 2.

In the present application, a toe or deep critical circular failuremechanism is assumed, and a total stress analysis of single rigid bodylimit equilibrium analytical method satisfying moment balance isadopted. As a result, an optimal minimum stability number N and criticalslip surface are successfully obtained based on the analytic solution ofstability number N of Hunter & Schuster (1968) by using a self-developedgenetic algorithm Python optimization program. The present applicationfurther proposed an analytic method of an equivalent unit weight basedon the obtained critical slip surface.

The equivalent problems shown in FIGS. 2 and 4 have the same slopegeometry and undrained strength profile and will be assumed to have thesame critical failure surface and factor of safety. The critical failuresurface may be determined by optimizing the minimum stability number Nfrom Equation (3), which is a function of β and M but not γ.Furthermore, the factor of safety in terms of moments is given by:

$\begin{matrix}{{FS} = \frac{M_{R}}{M_{O}}} & \left( {12 - 1} \right)\end{matrix}$where M_(R) and M₀ are given as:

$\begin{matrix}{M_{R} = {\frac{\rho{RH}^{2}}{2\sin\alpha\sin\lambda}\left\lbrack {{\cot\lambda} + {\alpha\left( {1 + {2M} - {\cot\alpha\cot\lambda}} \right)}} \right\rbrack}} & \left( {12 - 2} \right)\end{matrix}$ $\begin{matrix}{M_{O} = {{\frac{\gamma H^{3}}{12}\left\lbrack {1 - {2\cot^{2}\beta} + {3\cot\lambda\cot\beta} + {3\cot\alpha\cot\lambda} - {3\cot{\alpha cot}\beta}} \right\rbrack}.}} & (12)\end{matrix}$

From equations (12-2) and (12), it can be noted that the overturningmoment M₀, but not the resisting moment M_(R), is dependent on γ. For agiven critical failure surface, since M_(R) is the same in both cases,it is obvious that as long as M₀ is the same, the factor of safety willalso be unchanged.

In this method, the accuracy of the factor of safety is improved byensuring the equivalent principle of a constant overturning moment M₀when homogenizing the unit weight and taking the influence of the slopeinclination β on the unit weight into consideration. Therefore, a newset of equivalent unit weight chart suitable for a wide variety of waterlevel d_(w)/H conditions against different slope inclination β which canbe used in combination with a stability chart is proposed, whichprovides a new tool for convenient and accurate engineeringapplications.

Based on the above finding, the present application proposes a methodfor computing the factor of safety from equivalent unit weight for aslope, which includes the steps of optimizing stability number N andcalculating equivalent unit weight. In some embodiments, a method forcomputing the factor of safety of a slope includes the following steps.

Step 1: acquiring parameters of the slope including saturated unitweight of soil (γ_(sat)) and unit weight of water (γ_(w)), slopegeometry (β, H), soil strength (s_(u0), ρ, M) and external water level(d_(w)/H);

Step 2: formulating a formula of stability number N into a targetfunction and determining constrained conditions of variable geometricparameters α and λ in the target function;

Step 3: retrieving a combination of geometric parameters α and λ bycalling a genetic algorithm library through a Python program to obtain aminimum stability number N;

Step 4: determining a position of a critical slip surface according tothe retrieved combination of the variable geometric parameters α, λ andthe slope geometry acquired in Step 1;

Step 5: transforming the determined critical slip surface into acritical slip surface defined by a circular arc radius R of the criticalslip surface and horizontal and vertical coordinate values x_(c) andy_(c) of a circle center of the critical slip surface, and computingoverturning moments of slid masses above and below external water levelof the critical slip surface;

Step 6: resolving an equivalent unit weight γ_(equiv) according tooverturning moment equilibrium; and

Step 7: resolving the factor of safety from the obtained minimumstability number N, a gradient ρ of soil strength and the equivalentunit weight γ_(equiv) obtained in Step 6 according to the followingformula (1):

$\begin{matrix}{{FS} = {N{\frac{\rho}{\gamma}.}}} & (1)\end{matrix}$

In some embodiments, after determining a minimum stability number N anda critical slip surface via an optimization program, the critical slipsurface defined by α, λ and n is changed into that defined by R, x_(c)and y_(c). Based on three types of slope failure mechanisms and using anexternal water line and a horizontal line at the slope toe asinterfaces, the deep circle shown in FIG. 5A and the deep toe circleshown in FIG. 5B for a slip surface passing below the toe of the slopeare divided into three sections, and the shallow toe circle for a slipsurface without passing below the toe of the slope as shown in FIG. 5Cis divided into to two sections.

In some embodiments, the unit weight of ABHO is set as a saturated unitweight, and the unit weight of BDTH is set as a buoyant unit weight, andthe unit weight of DEFT is set as a buoyant unit weight, as shown inFIG. 6A. In some embodiments, the unit weight of ABHO is set as asaturated unit weight, the unit weight of BDTH is set as a buoyant unitweight, and the unit weight of DET is set as a buoyant unit weight, asshown in FIG. 7A. In some embodiments, the unit weight of ABHO is set asa saturated unit weight, and the unit weight of BTH is set as a buoyantunit weight, as shown in FIG. 8A.

In some embodiments, according to the principle of moment equilibrium,the sum of the overturning moments of ABHO, BDTH, and DEFT or DET slipmasses as shown in FIGS. 6A and 7A is equal to the overturning momentproduced by the whole slip mass having an equivalent unit weight asshown in FIGS. 6B and 7B, which gives the equation (21) of overturningmoment equilibrium:M ₁ +M ₂ +M ₃ =M ₀  (21)where M₁ denotes an overturning moment of the saturated unit weight slipmass ABHO above the external water level, M₂ denotes an overturningmoment of the buoyant unit weight slip mass BDTH between the externalwater level and the toe of the slope, M₃ denotes the overturning momentof the buoyant unit weight slip mass DEFT or DET below the toe of theslope, and M₀ denotes the total moment of the whole slip mass having anequivalent unit weight. In some further embodiments, M₃ is 0.

In some other embodiments, according to the principle of momentequilibrium, the sum of the overturning moments of ABHO and BTH slipmasses as shown in FIG. 8A is equal to the overturning moment producedby the whole slip mass having an equivalent unit weight as shown in FIG.8B, which gives the equation (8) of overturning moment equilibrium:M ₁ +M ₂ =M ₀  (8)

In some embodiments,M ₁=γ_(sat)∫_(−d) _(w) ⁰(x _(C) −x _(G))dA  (9)where γ_(sat) denotes a saturated unit weight of soil, x_(c) denotes thehorizontal coordinate value of the center of the circle corresponding tothe interface arc formed by the critical slip surface, x_(G) denotes thehorizontal coordinate value of the centroid of a thin horizontalintegral element of soil at a general depth y, d_(w) denotes the depthof water outside slope measured below the crest, and dA denotes theintegral element area.

In some embodiments,M ₂=γ′∫_(y) _(T) ^(−d) ^(w) (x _(C) −x _(G))dA  (10)where γ′ denotes buoyant unit weight of soil, and y_(T) denotes avertical coordinate of a slope toe.

In some embodiments,M ₀=γ_(equiv)∫_(y) _(T) ⁰(x _(C) −x _(G))dA  (11)where γ_(equiv) denotes the equivalent unit weight of soil.

Equation (11) is equal to equation (12) as deduced by Hunter & Schuster(1968)

$\begin{matrix}{M_{O} = {{\frac{\gamma H^{3}}{12}\left\lbrack {1 - {2\cot^{2}\beta} + {3\cot\lambda\cot\beta} + {3\cot\alpha\cot\lambda} - {3\cot{\alpha cot}\beta}} \right\rbrack}.}} & (12)\end{matrix}$where H denotes the height of the slope, γ denotes the unit weight of aslope having a uniform unit weight.

Therefore, it gives

$\begin{matrix}\begin{matrix}{M_{O} = {M_{1} + M_{2}}} \\{= {\gamma_{equiv}{\int_{y_{T}}^{0}{\left( {x_{C} - x_{G}} \right){dA}}}}} \\{= {\frac{\gamma H^{3}}{12}\left\lbrack {1 - {2\cot^{2}\beta} + {3\cot\lambda\cot\beta} + {3\cot\alpha\cot\lambda} - {3\cot{\alpha cot}\beta}} \right\rbrack}}\end{matrix} & (13)\end{matrix}$which reflects the equivalent relationship between the overturningmoment of partially submerged undrained slope as shown in FIG. 2 andthat of a uniform slope without external water surface and havingequivalent unit weight as shown in FIG. 4.

In some embodiments, in order to resolve the equations (9), (10) and(11), from a circular arc equation of a slip surface:

$\begin{matrix}{{x = {x_{C} - \sqrt{R^{2} - \left( {y - y_{C}} \right)^{2}}}},{and}} & (14)\end{matrix}$a slope equation:

$\begin{matrix}{x = {- \frac{y}{\tan\beta}}} & (15)\end{matrix}$it gives

$\begin{matrix}{x_{G} = {\frac{1}{2}\left( {x_{C} - \sqrt{R^{2} - \left( {y - y_{C}} \right)^{2}} - \frac{y}{\tan\beta}} \right)}} & (16)\end{matrix}$which further gives

$\begin{matrix}{{x_{C} - x_{G}} = {\frac{1}{2}\left\lbrack {\sqrt{R^{2} - \left( {y - y_{C}} \right)^{2}} + \left( {\frac{y}{\tan\beta} + x_{C}} \right)} \right\rbrack}} & (17)\end{matrix}$ $\begin{matrix}{{dA} = {\left\lbrack {\sqrt{R^{2} - \left( {y - y_{C}} \right)^{2}} + \left( {\frac{y}{\tan\beta} + x_{C}} \right)} \right\rbrack{dy}}} & (18)\end{matrix}$where for a deep toe circle and a shallow toe circle, the critical slipsurface can also be expressed as:

$\begin{matrix}{R = \frac{H}{2\sin\alpha\sin\lambda}} & (7)\end{matrix}$ $\begin{matrix}{x_{c} = {\frac{H}{\tan\beta} - {R{\sin\left( {\alpha - \lambda} \right)}}}} & \left( {7 - 1} \right)\end{matrix}$ $\begin{matrix}{{y_{c} = {{R\cos\left( {\alpha - \lambda} \right)} - H}},{and}} & \left( {7 - 2} \right)\end{matrix}$for a deep circle, the critical slip surface can also be expressed as:

$\begin{matrix}{R = \frac{H}{2\sin\alpha\sin\lambda}} & (7)\end{matrix}$ $\begin{matrix}{x_{c} = {\frac{H}{\tan\beta} - \left\lbrack {{R{\sin\left( {\alpha - \lambda} \right)}} - {nH}} \right\rbrack}} & \left( {7 - 3} \right)\end{matrix}$ $\begin{matrix}{y_{c} = {{R\cos\left( {\alpha - \lambda} \right)} - {H.}}} & \left( {7 - 4} \right)\end{matrix}$

Substituting equations (17) and (18) into equations (9)-(11) and theninto equation (13) provides an equivalent unit weight:

$\begin{matrix}{\gamma_{equiv} = {\frac{{\gamma_{sat}{\int_{- d_{W}}^{0}{\left( {x_{C} - x_{G}} \right){dA}}}} + {\gamma^{\prime}{\int_{y_{T}}^{- d_{W}}{\left( {x_{C} - x_{G}} \right){dA}}}}}{\int_{y_{T}}^{0}{\left( {x_{C} - x_{G}} \right){dA}}}.}} & (19)\end{matrix}$

According to the analytical solution of equivalent unit weight obtainedfrom the optimal critical slip surface, a verification is performedbelow by using an optimization program based on the analytical equation(1) and (3) regarding two aspects:

$\begin{matrix}{{{FS} = {N\frac{\rho}{\gamma}}},{and}} & (1)\end{matrix}$ $\begin{matrix}{N = {\frac{3}{\sin^{2}\alpha\sin^{2}\lambda}{\frac{\left\lbrack {{\cot\lambda} + {\alpha\left( {1 + {2M} - {\cot\alpha\cot\lambda}} \right)}} \right\rbrack}{\begin{matrix}\left( {1 - {2\cot^{2}\beta} + {3\cot\lambda\cot\beta} +} \right. \\\left. {{3\cot\alpha\cot\lambda} - {3\cot\alpha\cot\beta}} \right)\end{matrix}}.}}} & (3)\end{matrix}$

(a) The stability chart of undrained slope (ϕ=0) where strengthincreases linearly with depth was initially established by Hunter &Schuster (1968) regarding slopes having homogeneous unit weight, forexample, a submerged case or a case without water outside the slope.Therefore, the two cases are verified below, and the results are shownin FIG. 9 for comparison. For a slope inclination of 5°-30°, the resultobtained by Python optimization (dotted line in FIG. 9) is very similarto, in particular, slightly larger than, that (solid line in FIG. 9) inthe stability chart of Hunter & Schuster (1968). For a slope inclinationof 30°-90°, the result (dotted line in FIG. 9) obtained by Pythonoptimization is well consistent with that (solid line in FIG. 9) in thestability chart of Hunter & Schuster (1968).

(b) For a partially submerged undrained slope with external water level,taking the situation where γ_(sat)=20 kN/m³, M=2, and β=30° as anexample, the factors of safety obtained from the above equivalent unitweight analysis by the Python optimization program are verified againstthat obtained by finite element strength reduction method, as shown inFIG. 10. It can be seen from the result that the factors of safetyobtained by both methods are well consistent with each other, confirmingthat the analytical solution of equivalent unit weight by equivalentoverturning moment are correct.

For studying the effect of dimensionless strength gradient parameter Monequivalent unit weight γ_(equiv), the equivalent unit weight is furtherand comprehensively considered based on the following parameters, inorder to reveal the relevant variables on which the equivalent unitweight depends on and make the value range of the dimensionless strengthgradient parameter M and the slope inclination β consistent with thestability chart.

γ_(sat)=15 kN/m³, 16 kN/m³, 17 kN/m³, 18 kN/m³, 19 kN/m³, 20 kN/m³;

β=5°, 10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°, 90°;

M=0.0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0;

d_(w)/H=0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0.

A series of optimization calculations show the equivalent unit weightγ_(equiv) is primarily affected by γ_(sat), β and d_(w)/H. Furthermore,typical results for the case of γ_(sat)=16 kN/m³, d_(w)/H=0.5 withM=0.0, 0.75, 1.5, 2.0, illustrate the influence of M on the equivalentunit weight, as shown in FIG. 11. It is noted that all γ_(equiv) valueslie in a narrow band between M=0 and M=2.0 with the differences acrossthe entire range of β not exceeding 0.34 kN/m³. It is also noted thatthe γ_(av) value from simple averaging (equation 20 and FIG. 11) isalways less than even the lowest boundary values of γ_(equiv) by up to2.27 kN/m³ for higher values β.

$\begin{matrix}{\gamma_{av} = {\frac{{d_{w}\gamma_{zat}} + {\left( {H - d_{w}} \right)\gamma^{\prime}}}{H} = {\gamma_{sat} - {\left( {1 - \frac{d_{w}}{H}} \right)\gamma_{w}}}}} & (20)\end{matrix}$

Other results for the case of different γ_(sat) and d_(w)/H gave asimilar trend as clearly indicated by the different values of η for theγ_(equiv) and γ_(av) cases shown in FIG. 14. Thus, in order to obtainconservative values of the factor of safety and to neglect therelatively insignificant influence of M on the equivalent unit weight,it is reasonable to use the upper boundary values of γ_(equiv) inequation (1).

Further, for studying the influence of average unit weight on factor ofsafety, use of γ_(av) from equation (20) will be now compared withresults obtained using the equivalent unit weight γ_(equiv) as presentedin FIG. 14 and equation (22).

A typical set of results for the case of γ_(sat)=16 kN/m³ is shown inFIGS. 12 and 13. A saturated unit weight of γ_(sat)=16 kN/m³ is chosenbecause the smaller the saturated unit weight, the greater the impact ofthe average unit weight on the factor of safety.

It is obvious in FIG. 12 that the errors increase with increasing slopeangle due to the upper boundary values of γ_(equiv) moving further awayfrom the constant γ_(av) value given by equation (20). The differencesbetween the two are about 0.04˜1.55 kN/m³, 0.77˜2.4 kN/m³, 1.19˜2.06kN/m³, respectively, for d_(w)/H=0.2, 0.5, 0.7. Values of γ_(equiv) andγ_(av) are the same for d_(w)/H=0.0, 1.0.

For comparison, FS from the upper and the lower boundary value ofγ_(equiv) and the value of γ_(av) given by equation (20) against slopeangle are plotted for d_(w)/H=0.0, 0.2, 0.5, 0.7, 1.0, as shown in FIG.13. It is worth noting that the differences between FS from the upperand lower boundary values of γ_(equiv) is very small compared with thedifferences between FS from the upper boundary value of γ_(equiv) andthe γ_(av) which increase with increasing slope inclination. FS from theγ_(av) value given by equation (20) are 19.0%, 22.1% and 15.8% greaterthan that from the upper boundary value of γ_(equiv), respectively, ford_(w)/H=0.2, 0.5, 0.7. Similarly, with γ_(sat)=20 kN/m³, FS from theγ_(av) value given by equation (20) are 12.7%, 16.0% and 12.1% greaterthan that from the upper boundary value of γ_(equiv), respectively, forthe same values of d_(w)/H. Validations of the factor of safety byfinite element strength reduction reveal that the charts presented inthis present application using γ_(equiv) lead to lower and moreconservative estimates of the factors of safety while those obtainedusing the simple weighted average approach lead to higher andunconservative estimates for almost all cases.

In summary, equation (20) ignores the influence of the slope angle andthe changing overturning moment, leading to an underestimation of theunit weight and overestimation of the factor of safety.

For the purpose of obtaining a more accurate factor of safety FS, anequivalent unit weight chart which can be used at a wide variety ofwater levels d_(w)/H for different saturated unit weight γ_(sat) andslope inclinations β in combination with the stability chart developedby Hunter & Schuster (1968) is proposed in the present application, inwhich the upper boundary values of the equivalent unit weight γ_(equiv)under a variety of M conditions are acquired, ignoring the very littleinfluence of Mon γ_(equiv). Such an equivalent unit weight chart, arepresented in FIG. 14 in the form of a dimensionless parameter Θ=f (β,d_(w)/H), whereγ_(equiv)=γ_(sat)−ηγ_(w)  (22)

The η values needed in equation (22) are given by the solid black linesin FIG. 14, but for comparison, the present application has alsoincluded as horizontal dotted lines, the equivalent η given by thesimple averaging approach from equation (20). In the simple averagingapproach, η=(1−d_(w)/H) and does not depend on β. It should be notedthat η from the simple averaging approach for a given value of d_(w)/H(dotted lines), is nearly always higher than that from the equivalentunit weight approach (solid lines) indicating that γ_(equiv)>γ_(av). Ahigher unit weight results in a lower FS, hence the equivalent unitweight is more conservative than the simple averaged unit weight.

The following example is provided merely for the purpose of betterunderstanding the present application for those skilled in the art, notintended to be considered as limiting the scope of protection of thepresent application in any way.

Computing Example

Consider the partially submerged undrained slope with geometry, waterdepth and strength profile as shown in FIG. 15.

From FIG. 9, with M=1 and β=30°, N=12.2 and from FIG. 14 with β=30° and

${\frac{d_{w}}{H} = 0.5},$η=0.335 hence γ=γ_(equiv)=γ_(sat)−ηγ_(w)=16−0.335×9.81=kN/m³

Finally, the factor of safety is given from equation (1) as:

${FS} = {{N\frac{\rho}{\gamma}} = {{12.2 \times \frac{1}{12.7}} = 0.96}}$

In comparison, from equation (20) or FIG. 14,

${\eta = {{1 - \frac{d_{w}}{H}} = 0.5}}{\gamma_{av} = {{\gamma_{sat} - {\eta\gamma}_{w}} = {{16 - {0.5 \times 9.81}} = {11.1\frac{kN}{m^{3}}}}}}$and

${FS} = {{N\frac{\rho}{\gamma}} = {{12.2 \times \frac{1}{11.1}} = 1.1}}$

The relative error from the average unit weight approach is thereforegiven by:

${Error} = {{\frac{1.1 = 0.96}{0.96} \times 100\%} = {15\%}}$

As a check, the factor of safety from finite element strength reductionwas given as 0.97.

It can be seen that, the factor of safety calculated from the equivalentunit weight according to the present application was 0.96, indicatingthat the slope was in an unstable state. In contrast, the factor ofsafety obtained from the average unit weight given by equation (20) was1.10 which is larger than 1, indicating that the slope is in a stablestate. Therefore, the factor of safety obtained from the average unitweight given by equation (20) had an error of 15% in comparison thatobtained according to the present application, which was anoverestimated factor of safety, leading to unsafe estimation.

What is claimed is:
 1. A method for constructing a slope of a groundsurface, adapted to determine a factor of safety FS of the slope of theground surface for determining whether the slope of the ground surfaceis in a stable state and construct the slope of the ground surface in aslope engineering, and comprising the following steps: Step 1: acquiringparameters of the slope of the ground surface including unit weights ofsoil, slope geometry, soil strength and external water level; Step 2:formulating a formula of stability number N into a target function anddetermining constrained conditions of variable geometric parameters inthe target function; Step 3: retrieving a combination of the variablegeometric parameters by calling a genetic algorithm library through aPython program to obtain a minimum stability number N_(min); Step 4:determining a critical slip surface defined by a position according tothe retrieved combination of the variable geometric parameters and theslope geometry acquired in Step 1; Step 5: transforming the determinedcritical slip surface into a critical slip surface defined by a circulararc radius R of the critical slip surface and horizontal and verticalcoordinate values x_(c) and y_(c) of a circle center of the criticalslip surface, and determining overturning moments of slid masses aboveand below external water level of the critical slip surface; Step 6:determining an equivalent unit weight γ_(equiv) according to overturningmoment equilibrium; and Step 7: determining the factor of safety FS fromthe obtained minimum stability number N_(min), a gradient ρ of soilstrength and the equivalent unit weight γ_(equiv) obtained in Step 6according to the following formula (1): $\begin{matrix}{{{FS} = {N_{\min}\frac{\rho}{\gamma}}},} & (1)\end{matrix}$ where γ is equal to the equivalent unit weight γ_(equiv),and when the obtained FS is larger than 1.0, the slope of the groundsurface is constructed.
 2. The method according to claim 1, wherein, inStep 1, the unit weights of soil comprise a saturated unit weight ofsoil, a unit weight of water, an assigned saturated unit weight of soilabove external water, and an assigned buoyant unit weigh of soil belowexternal water, the slope geometry comprises a slope angle and a heightof the slope of the ground surface, the soil strength comprises anundrained strength of the slope of the ground surface given by theequation:s _(u)(z)=s _(u0) +ρz  (2) where s_(u0) is the soil strength at crestlevel (z=0) and ρ is the gradient of soil strength, and the externalwater level is defined as d_(w)/H here d_(w) denotes a depth of waterlevel outside the slope from top of the slope of the ground surface, andH denotes the height of the slope of the ground surface.
 3. The methodaccording to claim 2, wherein, in Step 2, the target function for acircular arc failure mechanism is expressed as: $\begin{matrix}{{N = {\frac{3}{\sin^{2}\alpha\sin^{2}\lambda}\frac{\left\lbrack {{\cot\lambda} + {\alpha\left( {1 + {2M} - {\cot\alpha\cot\lambda}} \right)}} \right\rbrack}{\begin{matrix}\left( {1 - {2\cot^{2}\beta} + {3\cot\lambda\cot\beta} +} \right. \\\left. {{3\cot\alpha\cot\lambda} - {3\cot\alpha\cot\beta}} \right)\end{matrix}}}},} & (3)\end{matrix}$ where α denotes half of a central angle of a circular arcslip surface, λ denotes an angle between a chord line of the circulararc slip surface and a horizontal plane, β denotes an angle of the slopeof the ground surface, and M denotes a dimensionless strength gradientparameter defined as: $\begin{matrix}{M = {\frac{h}{H}\frac{\gamma_{w}}{\gamma^{\prime}}}} & (4)\end{matrix}$ where h denotes a specific water level in the slope of theground surface, γ_(w) denotes the unit weight of water, and γ′ denotes abuoyant unit weight.
 4. The method according to claim 3, wherein, inStep 2, M is further defined as: $\begin{matrix}{M = {\frac{H_{0}}{H} = \frac{s_{u0}}{\rho H}}} & (5)\end{matrix}$ where H₀ denotes an intercept of a strength line.
 5. Themethod according to claim 4, wherein, in Step 2, the variable geometricparameters comprise α and λ, and the constrained conditions comprise: α:[0, 90°] and λ: [0, β].
 6. The method according to claim 5, wherein, inStep 4, the position of the critical slip surface is defined by$\begin{matrix}{n = {\frac{1}{2}\left( {{\cot\lambda} - {\cot\alpha} - {\cot\beta}} \right)}} & (6)\end{matrix}$ where a dimensionless value n denotes a measure of adistance the critical slip surface outcrops beyond a toe of the slope ofthe ground surface; and, when n is zero or negative, a critical circlepasses through the toe, and when n is positive, the critical circleexists below the toe.
 7. The method according to claim 6, wherein, inStep 5, for a deep toe circle or a shallow toe circle, transforming thedetermined critical slip surface into the critical slip surface definedby R, x_(c) and y_(c) is performed by $\begin{matrix}{R = \frac{H}{2\sin\alpha\sin\lambda}} & (7)\end{matrix}$ $\begin{matrix}{x_{c} = {\frac{H}{\tan\beta} - {R{\sin\left( {\alpha - \lambda} \right)}{and}}}} & \left( {7 - 1} \right)\end{matrix}$ $\begin{matrix}{y_{c} = {{R{\cos\left( {\alpha - \lambda} \right)}} - {H.}}} & \left( {7 - 2} \right)\end{matrix}$
 8. The method according to claim 7, wherein, in Step 5,for a deep circle, transforming the determined critical slip surfaceinto the critical slip surface defined by R, x_(c) and y_(c) isperformed by $\begin{matrix}{R = \frac{H}{2\sin\alpha\sin\lambda}} & (7)\end{matrix}$ $\begin{matrix}{x_{c} = {\frac{H}{\tan\beta} - {\left\lbrack {{R{\sin\left( {\alpha - \lambda} \right)}} - {nH}} \right\rbrack{and}}}} & \left( {7 - 3} \right)\end{matrix}$ $\begin{matrix}{y_{c} = {{R{\cos\left( {\alpha - \lambda} \right)}} - {H.}}} & \left( {7 - 4} \right)\end{matrix}$
 9. The method according to claim 8, wherein, in Step 6,the overturning moment equilibrium is expressed as:M ₁ +M ₂ =M ₀  (8) where M₀ denotes a total moment of a whole slip masshaving an equivalent unit weight, M₁ denotes an overturning moment ofsaturated unit weight of slip mass above the external water level, andM₂ denotes an overturning moment of the buoyant unit weight of slip massbetween the external water level and the toe of the slope of the groundsurface.
 10. The method according to claim 9, wherein, in Step 6, M₁ isdefined as:M ₁=γ_(sat)∫_(−d) _(w) ⁰(x _(C) −x _(G))dA  (9) where γ_(sat) denotessaturated unit eight of soil, x_(G) denotes a horizontal coordinatevalue of a centroid of a thin horizontal integral element of soil at ageneral depth y, d_(w) denotes a depth of water outside the slope of theground surface measured below a crest, and dA denotes an integralelement area.
 11. The method according to claim 10, wherein, in Step 6,M₂ is defined as:M ₂=γ′∫_(y) _(T) ^(−d) ^(w) (x _(C) −x _(G))dA  (10) where γ′ denotesthe buoyant unit weight, and y_(T) denotes a vertical coordinate of thetoe of the slope of the ground surface.
 12. The method according toclaim 11, wherein, in Step 6,M ₀=γ_(equiv)∫_(y) _(T) ⁰(x _(C) −x _(G))dA  (11).
 13. The methodaccording to claim 12, wherein, in Step 6, the overturning momentequilibrium is further expressed as: $\begin{matrix}\begin{matrix}{M_{o} = {M_{1} + M_{2}}} \\{= {\gamma_{equiv}{\int_{y_{T}}^{0}{\left( {x_{C} - x_{G}} \right){dA}}}}} \\{= {{\frac{\gamma H^{3}}{12}\left\lbrack {1 - {2\cot^{2}\beta} + {3\cot\lambda\cot\beta} + {3\cot\alpha\cot\lambda} - {3\cot\alpha\cot\beta}} \right\rbrack}.}}\end{matrix} & (13)\end{matrix}$
 14. The method according to claim 13, wherein, in Step 7,the equivalent unit weight γ_(equiv) is obtained from an equation (22)in which η is a dimensionless value obtainable from an equivalent unitweight diagram:γ_(equiv)=γ_(sat)−ηγ_(w)  (22).